## 3.1 Parallax The distance in pc of an object is equal to 1 / parallax angle (small angles only). Equation is then: #### (d/pc) = 1/($\varpi$ / arcsec) the uncertainty in distance: #### $\Delta$ $\varpi$ / $\varpi$ = $\Delta$ d / d so : #### $\Delta$ d = $\Delta$ $\varpi$ / $\varpi^2$ ## 3.2 Gaia and big data angular resolution is 10 $\mu$as eg ## 3.3 Standaard candles A [[../Glossary/standard candle]] is used to determine the distance to the object in question. The equation depicting the relation between distance and magnitude is: #### M=m-5 log(d/pc) +5-A - ### pulsating stars - [[../Glossary/RR Lyrae stars]], *all* LL Lyrae stars have a M$_v$ = 0.75 . This is an average but workable. [[../Assets/example 3.5]] is an example of a distance calculation. - because of their low luminosity (+0.75) they can't be seen at great distances. Furthest sofar was in M31. - [[../Glossary/Cepheids]] , The equation to determine the absolute magnitude is : - M$_v$ = -2.43 log (P/day) - 1.62 - Cepheid variable stars are intrinsically very luminous: their absolute magnitudes of around −1.5 to −6.5 correspond to between about 300 and 30 000 times the luminosity of the Sun. Therefore, they may be identified in quite distant galaxies. Python programs to calculate the parameters [[../Python programs/Pulsating star calculations]] ## 3.4 The Cosmic Distance Ladder ![[../Assets/Screenshot 2023-10-25 at 16.06.21.png]] - [[../Glossary/Type 1a supernova]]. Because the critical mass at which white dwarfs will explode is the _same_ for all [[../Glossary/white dwarf star]], they are all likely to be identical at this point, and in principle they will all therefore explode with the same luminosity. As a result, they can act as standard candles. It’s actually a little more complicated than this because the light curves of individual Type Ia supernovae have slightly different peak intensities and different rates of decline over the course of a month or so. Type Ia supernovae with fainter absolute magnitudes at their peak fade away more rapidly. The required stretch factor needed for any rate of decline has been worked out, so by correcting an individual light curve in this way, its apparent peak magnitude can be compared with the absolute peak magnitude of about *M$_B$= M$_V$ = -19.6 magnitude*. The absolute magnitude of Type Ia supernovae may be calibrated using supernovae in relatively nearby galaxies, whose distances are measured using Cepheid variables. It may then be used to measure the distances to even more distant galaxies, out to about 1 Gpc. A typical galaxy may exhibit roughly one Type Ia supernova per century. - [[../Glossary/The Hubble–Lemaître law]] . This is a relationship between the apparent speed at which a galaxy is receding from us and its distance away, which was first noticed by Belgian astronomer Georges Lemaître and American astronomer Edwin Hubble in the 1920s. They observed that the more distant a galaxy is, the faster it appears to be moving. We may write this relationship as: #### v = H$_0$ x d {H$_0$ is Hubble constant = 70 km s$^{-1}$ Mpc$^{-1}$ } where *v* is the galaxy’s apparent speed of recession, *d* is its distance away, and *$H_0$* is the Hubble constant, which has a value of about 70 km s$^{-1}$ Mpc$^{-1}$ For every Mpc increase in distance, the recession speed appears to increase by 70 km s−1. The apparent speed of recession of a galaxy may be measured from its spectrum and is quantified by a number known as the cosmological redshift, represented by *z*. At low speeds (less than about 20% of the speed of light, ), redshift is simply the speed of the galaxy divided by the speed of light (3.00 × 10$^5$ km s$^{-1}$): #### z = v/c [[Astronomy overview.canvas|Astronomy overview]] [[Docent/S284 Astronomy]] [[Part 2 Spectral lines]]